The purpose of this research is to examine the statistical inference for the Lindley distribution using step-stress partially accelerated life testing (SS-PALT) and an adaptive Type I progressive hybrid censoring scheme (AT-I PHCS). To improve the analysis of lifetime observations, the suggested method combines adaptive censoring processes with acceleration mechanisms. Both maximum likelihood and Bayesian approaches are used to estimate the acceleration factor and unknown distribution parameters. The asymptotic properties of the maximum likelihood estimators are derived, upon which approximate confidence intervals are built. In the Bayesian context, estimation is performed under the squared error loss function, while posterior summaries and credible intervals are obtained using Markov chain Monte Carlo (MCMC) methods. An extensive simulation study is carried out to evaluate the performance of the suggested estimators across different censoring schemes and acceleration conditions. Additionally, optimal design considerations are taken into account to determine efficient sampling plans within the SS-PALT setting. The practical relevance of the developed methodology is demonstrated through an application to real data, confirming its effectiveness in fitting lifetime data collected from accelerated experiments. Overall, the findings indicate that the combination of adaptive censoring strategies with partially accelerated life testing can lead to improved estimation accuracy and greater experimental efficiency.

Unified hybrid censored sampling results from the need to effectively evaluate survival data impacted by different censoring techniques, enabling researchers to use partial information and lower bias in parameter estimates. This study employs unified hybrid censored samples to examine the maximum likelihood and Bayesian estimation methods for unknown parameters of the Lindley distribution, encompassing the reliability function, hazard rate function, cumulative hazard function, and mean residual life. Initially, we estimate the model parameters via the maximum likelihood method. We also examine bootstrapped confidence intervals and asymptotic confidence intervals for unknown parameters of the Lindley distribution. We also derive the confidence interval for the mean residual life, the hazard rate function, the cumulative hazard function, and the reliability function. Moreover, using flexible gamma priors for parameters combined with a non-informative prior yields Bayes estimators based on the principles of squared error, linear exponential, and general entropy loss functions. Consequently, we identify the corresponding highest posterior density credible intervals for unknown parameters, the reliability function, the hazard rate function, the cumulative hazard function, and the mean residual lifetime. Using numerous criteria, Monte Carlo simulations help evaluate the accuracy of the given estimations. We also assess sampling techniques across different rival censoring systems, including bootstrap and censored samples. Ultimately, we investigated a medical data set to show the useful value of the suggested approaches. The numerical results confirm the efficiency of our proposed approaches.